紧电阻空间上偏微分方程的逼近
Approximation of partial differential equations on compact resistance spaces
摘要：
考虑电阻空间上的线性偏微分方程，这些方程在二次型的意义下是一致椭圆和抛物的，并且涉及抽象的梯度和发散项。我们的主要兴趣是为其独特的解决方案提供图和度量图近似。对于单个紧致电阻空间上的不同系数的方程族，只要系数有界，我们证明了解关于空间上的一致收敛性具有聚点.如果在一个方程序列中，系数适当地收敛，则解沿子序列一致收敛。对于局部电阻形式的特殊情况下，我们还考虑了电阻空间序列从内部逼近的局部电阻集。在适当的假设下，近似空间上方程解的线性化的系数（扩张）沿着目标方程解的一个子序列累积，甚至一致收敛于目标方程解的一个子序列。结果涵盖离散和度量图近似，两者都进行了讨论。
英文摘要：
We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.
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